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7/23/2019 Neural-network Applications
http://slidepdf.com/reader/full/neural-network-applications 1/11
ergamon
S0952-1976(96)00021-8
Engng App l ic Ar t i f I n te ll Vol. 9, No. 3, pp. 309-319, 1996
Copyright ~ 1996 Elsevier Science Ltd
Printed in Gr eat Britain. All rights reserved
0952-1976/96 $15.00 + 0.00
ontr ibu ted Paper
N e u r a l n e t w o r k A p p l i ca t io n s i n P r e d ic t in g M o m e n t c u r v a t u r e
P a r a m e t e r s f r o m E x p e r i m e n t a l D a t a
MANSOUR NASSER JADID
King Faisal University, K ingdom o f Saudi Arabia
DANIEL R. FAIRBAIRN
University of Edinburgh, Scotland
Received October 1994; in rev ised for m February 1996)
The obje ct ive o f th is s tudy is to demonstrate a concept and a methodology, rather than to bui ld a
fu l l- sca le knowledge-based sys t em m odel by incorpora ting m os t o f the fundam enta l aspec ts o f a
neura l ne twork to so lve the complex non- l inear mapp ing for a beam-co lumn jo in t . Th i s paper
presents the conce pt o f paral lel d is tr ibuted processing base learning in arti ficial neural netw orks , in
assis ting wi th experime ntal evidence to predic t mo me nt-cur vatur e parameters that are usual ly
acco mplishe d solely by experime ntal work . General ly, i t ma y be possible to ident i fy certain
parameters , and a l low the neura l ne twork to deve lop the model , t hus accoun t ing for the observed
beha viour wi thou t relying on a part icular algori thm, but depending ent irely on the manipulat ion of
num erical data. Cop yright 0 1996 Elsevier Science Lt d
K e y w o r d s : N e u r a l n e t w o r k s , b a c k p r o p a g a t i o n , s t ru c t u ra l , r e in f o rc e m e n ,t , b e a m s , m o m e n t -
c u r v a t u r e , l o a d - d e fl e c ti o n , b e a m - c o l u m n j o i n ts .
1 . I N T R O D U C T I O N
C o n v e n t i o n a l m e t h o d s b a s e d o n t h e v o n N e u m a n n
pr og r a m m i ng i n s t r u c ti on ha v e e s t a b l i s he d t he b a s i s f o r
a l l c o m p u t a t i o n . H o w e v e r , t h i s m e t h o d i s c o m p u -
t a t i ona l l y i n t e ns i ve a nd s e r i a l i n i m p l e m e n t a t i on , s o
t h a t i t d o e s n o t p a r a l le l t h e h u m a n w a y o f th i n k in g a n d
a b s t ra c t i o n . T h e s y m b o l i c - b a s e d l e a rn i n g a p p r o a c h
r e d u c e s t h e c o m p u t a t i o n a l p r o c e s s i n th e p r e s e n c e o f a
h u g e a m o u n t o f d a t a a n d h a s a w e a k n e s s in k n o w l e d g e
r e p r e s e n t a t i on . S ym b o l i c l og i c - b a s e i m p l e m e n t a t i on i s
e x t r e m e l y d i ff ic u l t, pa r t i c u l a r ly i n t he a r e a o f i n te ns i ve
m a t h e m a t i c a l c o m p u t a t i o n . N e u r a l - n e t w o r k f i e l d s
p r o v i d e a c o m p u t a t i o n a l p a r a d i g m c h a l le n g e w h i c h h a s
r e s u lt e d f r o m f o u r d e c a d e s o f i n te n s iv e r e s e a r c h a n d
i n v e s t m e n t . T h e f u n c t io n o f t h e h u m a n b r a i n c o n t a in s
m a ny f e a t u r e s t ha t c a n b e s i m u l a t e d i n a m a c h i ne t o
pe r f o r m c e r t a i n t a s k s t ha t a r e d i f f i c u l t t o a c h i e ve b y
c o n v e n t i o n a l m e t h o d s a n d a sy m b o l i c a p p r o a c h . O n e o f
orrespondence shou ld b e s e n t t o : D r D . R . F a i r b a i r n , D e p a r t m e n t
o f C i vi l a n d E n v i r o n m e n t a l E n g i n e e r i n g , T h e U n i v e r s i t y o f
E d i n b u r g h , C r e w B u i l d i n g , T h e K i n g ' s B u i l d i n g s , E d i n b u r g h E H 9
3 J N , S c o t l a n d . E m a i l : e n e i 0 1 @ c a s t l e . e d . a c . u k .
309
t he p r i m a r y i n t e n t i ons o f the d i s t r i b u t e d c om m u n i t y is
t o d e s i g n n e w h a r d w a r e a n d s o f t w a r e o n a c o m p u t e r
t ha t c a n s i m u l a t e hu m a n t h i nk i ng .
2 . G E N E R A L V I E W O F A P P L I C A T I O N S T O C I V I L
E N G I N E E R I N G
Inte res t in l ea rning in the f i e ld of c iv i l engineer ing
da t e s a s f a r b a c k a s 1966 , w h e n S p i l le r s I pu b l i s he d a
pa pe r e n t i t l e d " A r t i f i c i a l I n t e l l i ge nc e a nd S t r u c t u r a l
D e s i g n " . I t w a s a n a t t e m p t t o d e m o n s t r a t e t h e m o s t
e l e m e n t a r y l e a r n i ng c a pa b i l i t y i n s t r u c t u r a l de s i gn . H i s
p r i m a r y i n t e r e s t w a s t o a pp l y t o s t r u c t u r a l de s i gn , a nd
h i s m a i n c ons i de r a t i on w a s t o de m ons t r a t e t he pos s i b i -
l i t y o f u s i ng e x a m pl e s t o ge ne r a t e r u l e s . A s he
s u gge s t e d :
" E x a m p l e s p l a y a n i m p o r t a n t p a r t i n te a c h i n g a n d
l e a r n i ng i n hu m a ns a nd i t i s na t u r a l t o a s k w ha t
pa r t t he y s hou l d p l a y i n t he f o r m a l a da p t i ve
s ys t e m . B e c a u s e s t u de n t s a r e t a u gh t u s i ng e x a m -
p l e s , sh o u l d c o m p u t e r s b e t a u g h t t h r o u g h t h e u s e
o f e x am p l e s , a n d h o w ? A g a i n , t h e a n s w e r i s n o t
n o w a v a i l a b le b u t p e r h a p s e x a m p l e s m a y b e u s e d
a s c o m p l e t e s e t s o f i n d e p e n d e n t v e c t o r s a r e n o w
u s e d t o r e p r e s e n t f u n c t io n s . "
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3 10 M A N S O U R N A S S E R J A D I D a n d D A N I E L R . F A I R B A I R N : M O M E N T - C U R V A T U R E
T hi s s how s a n e a r l y a t t e m p t b y S p i ll e r s z t o u s e
e x a m p l e s a s a m e a n o f ge ne r a t i ng r u l e s, a nd c on f i rm s
the f ac t tha t wha t i s ava i l ab le today in l ea rning appl i -
c a t i ons i s m e r e l y a n i l lu s t ra t i on o f w ha t w a s s a i d b e f o r e .
L e a r n i ng f r om b r i dge s t r u c t u r a l fa i l u re w a s p r o pos e d
b y S t one , 2 w h e r e b y a h i e r a r c h ic a l k now l e dge - b a s e
c ou l d b e de ve l ope d t o p r e d i c a t e b r i dge f a i l u r e f r om a
s t u dy o f h i s to r i c a l da t a . T he m e t hod o f le a r n i ng
e m p l o y e d w a s b a s e d o n a n a l g o r i t h m d e v e l o p e d b y
N or r i s e t aL T he r oo t s o f t h is l e a rn i ng t e c hn i q u e
e m e r ge d f r om t he m e d i c a l d i a gnos is fi e ld w h e r e t he
r e l a t i on b e t w e e n t he s ym pt om s a nd t he d i s e a s e i s
ob s e r ve d f o r a nu m b e r o f pa t i e n t c a s e h i s to r i e s . T he
l e a r n i ng p r oc e s s i nvo l ve s t w o pha s e s : a
d i s c r i m i n a t i o n
a na l ys i s t ha t i s k now n a s t he s e r i a l a pp r oa c h , a nd t he
c o n n e c t i v i t y a l go r i t hm w h i c h a dop t s a pa r a l l e l
a pp r oa c h . I n t he d i s c r im i na t i on c a s e , a s e a r c h is c a r r i e d
ou t f o r a s i ng l e f e a t u r e , w h i l e t he c onne c t i v i t y a l go r -
i t hm s e a r c h i s f o r a g r ou p o f fe a t u r e s . T h e p r e s e nc e o r
a b s e nc e o f s u c h a f e a t u r e o r f e a t u r e s , i nd i c a t e s t ha t a n
e l e m e n t o r e l e m e n t s b e l ong t o one o r a no t he r c l a s s . I n
c a s e s o f u n c e r ta i n t y e n c o u n t e r , t h e m e t h o d o f s u p p o r t
logic i s used .
T he a pp l i c a t i on o f ne u r a l ne t w or k s t o s tr u c t u r a l
a na l y s i s w a s c a r r i e d o u t b y J a d i d a nd F a i r b a i r n 4 i n t he
f o r m o f a da p t i ve fi n it e e l e m e n t m e s h ge ne r a t i on . T he
s t u d y c o n c e n t r a t e d o n t h e p r o c e s s o f a d a p t i v e r e -
m e s h i ng o f a n i de a l i z e d s q u a r e s ha pe a n d a n i nd i v i du a l
t r ia ng l e b y u s ing t r i a ngu l a r e l e m e n t s . S u pe r v i s e d t r a i n -
i ng w a s u s e d i n t he a pp l i c a t ion o f the b a c k - p r op a ga t i on
l e a r n i ng a l go r i t hm w h i c h de a l s w i th t he p r ob l e m o f r e -
meshing s t ruc tu ra l e l ements in a s t ruc tu ra l ana lys i s .
T h e m a i n o b j e c t i v e o f th e a u t h o r s ' s t u d y w a s t o d e m o n -
s t r a t e t he a b i l i ty o f ne u r a l n e t w or k s t o e m p l oy r e - m e s h
s t r u c t u r a l e l e m e n t s w i t hou t u s i ng nu m e r i c a l ly i n t e ns i ve
c o m p u t a t i o n s . T h e f u n d a m e n t a l r e q u i r e m e n t w a s t h e
s e l e c t ion o f a f e a s i b l e a nd a p p r op r i a t e d om a i n t o ge ne r -
a t e t r a i n i ng a nd t e s t da t a .
A m o r e c o m p r e h e n s i v e r e s e a rc h w o r k u s in g n e u r a l-
n e t w o r k t e c h n i qu e s w a s u n d e r t a k e n b y J a d i d 5 t o s t u d y
p r e v i o u s e x p e r i m e n t a l s t u d i e s o f b e a m - c o l u m n j o i n t
pa r a m e t e r s f r om a d i f f e r e n t a ng l e , a nd e s t a b l i s h a
on ept a n d m e t h o d o l o g y t h a t w o u l d p r o v i d e r a p i d a n d
e c onom i c b e ne f i t s f o r f u t u r e e x pe r i m e n t a l r e s e a r c h .
3 . F O R M U L A T I O N O F T H E B A C K P R O P A G A T I O N
A L G O R I T H M
A r t i fi c ia l ne u r a l n e t w or k s a r e i n s p i r e d b y hu m a n
b i o l og ic a l ne u r a l ne t w or k s , w he r e b y t he y c a p t u r e t he
b r a i ny f u nc t i on m a n i pu l a t i on t o a pp r oa c h a s pe c i f i c
p r ob l e m b y a pp l y ing c e r t a i n r u l e s to a c h i e ve r e a s ona b l e
resu l t s . The s tudy of a r ti fi ci al neura l ne two rks i s
f ou nde d on a s e m i - e m pi r i c a l b a s e t o m ode l t he b e -
ha v i ou r o f t he b i o l og i c a l ne r ve c e ll s t ru c t u r e . T he
p r oc e s s i ng e l e m e n t i n a n a r ti fi c ia l ne t w o r k i s a na l ogou s
t o t he ne r ve c e l l i n t he hu m a n b r a i n . T he b r a i n i s
c om pos e d o f de ns e ne r ve c e l l s w h i c h a r e h i gh l y
i n t e r c onn e c t e d a nd e s t i m a t e d t o t o t a l 100 b i ll i on ( ne u r -
ons ) o f d i f f e r e n t t ype s , w h i c h a r e c ons t a n t l y s e nd i ng
a nd r e c e i v i ng m e s s a ge s . T he s e ne r ve c e l l s a r e f u nda -
m e n t a l e l e m e n t s t o t he c e n t r a l ne r vou s s ys t e m , a nd
de t e r m i ne a ny a c t i on w h i c h i s t a k e n .
B a c k p r o p a g a t i o n n e t w o r k s a r e c o n s i d e r e d t o b e t h e
m os t r e l i a b l e a nd m os t a pp l i c a b l e o f al l. O ne s u r ve y ha s
s how n t ha t a b ou t 80% o f al l a pp l i c a t i ons u s e d b a c k -
p r opa ga t i on , du e t o t he m a t he m a t i c a l de s i gn o f l e a r n -
i ng c om pl e x non l i ne a r r e l a t i ons h i ps . E ve n t he i npu t
da t a i s l e s s p r e c i s e o r no i s y . A b a c k p r opa ga t i on
ne t w or k ha s t he a b i l i t y t o m i n i m i z e t he m e a n - s q u a r e d
e r r o r b y a pp l y i ng a g r a d i e n t - de s c e n t a l go r i t hm t ha t
f o l l ow s t he g r a d i e n t e r r o r c u r ve dow nw a r d a c r os s a ll
t he i npu t pa t te r n s . T h i s i s m a t he m a t i c a l l y c om p u t e d b y
t a k i ng t he pa r t i a l de r i va t i ve s o f the e r r o r w i t h r e s pe c t
t o t he w e i gh t s . A s u pe r v i s e d l e a r n i ng t e c hn i q u e i s
e s s e n t ia l i n the l e a r n i ng p r oc e s s , d u e t o t he p r e s e nc e o f
i n p u t a n d o u t p u t d a t a w h i c h e n s u r e s t h e d e v e l o p m e n t
o f an i n t e r na l m o de l t ha t de s c r i b e s t he ob j e c t i ve
r e q u i r e m e n t s .
T h e b a c k p r o p a g a t i o n a l g o ri th m i n v o lv e s a f o r w a r d
p r opa ga t i on s t a r t , w he n a s e t o f inpu t pa t t e r n s i s
p r e s e n t e d t o t h e n e t w o r k , a n d t h e b a c k w a r d e r r o r
a c t i va ti on b e g i ns a t t he ou t p u t l a ye r w he n e r r o r s p r opa -
ga t e t h r ou gh t he i n t e r m e d i a t e l a ye r s t ow a r d t he i npu t
l ay e r . T h e p r o c e s s o f f o r w a r d a n d b a c k w a r d p r o p a -
ga t i on c on t i nu e s u n t i l t he e r r o r i s r e du c e d t o a n a c c e p -
tab le l eve l , or has run for a spec i f i ed t ime . F igure 1
s how s a s i m p l i fi e d s ing l e p r oc e s s i ng e l e m e n t o f a b a c k -
p r o p a g a t i o n n e t w o r k w i th i t s s u m m a t i o n a n d a c t iv a t io n
f u nc t i ons w i t h i n a t yp i c a l b a c k p r opa ga t i on ne t w or k o f
t h r e e l a ye r s , e a c h o f w h i c h i s c on ne c t e d t o t he p r oc e s s -
i ng e l e m e n t s i n t he ne x t l a ye r .
4 . F U N D A M E N T A L M A T H E M A T I C A L
F O R M U L A T I O N
T h e f u n d a m e n t a l m a t h e m a t i ca l f o r m u l a t i o n o f th e
b a c k p r o p a g a t i o n n e t w o r k r e q u i r es e a c h p r o c e s s i n g e le -
m e n t t o p e r f o r m f o u r m a i n s t e ps :
1 . I npu t c onne c t i ons , w h i c h a r e a na l ogou s t o t he
s y n a p s e s , r e c e i v e i n f o r m a t i o n f r o m o t h e r p r o -
c e s s i ng e l e m e n t s o r s t a r t w i t h k now n i npu t
da t a .
2 . A s u m m a t i on f u nc t i on , w h i c h invo l ve s the a c t i-
va t i on o f e a c h p r oc e s s i ng e l e m e n t w i t h i t s
weight .
3 . A t h r e s ho l d f u nc t i on , w h i c h i s a p r oc e s s o f
c onve r t i ng t he s u m m a t i on i npu t a c t i va t i on da t a
t o a n ou t pu t a c t i va t i on da t a b y u s i ng a s pe c i f i c
f u nc t i on .
4 . O u t pu t p r oc e s s i ng e l e m e n t s , w h i c h r e s e m b l e
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MANSOUR NASSER JADID and DANIEL R. FAIRBAIRN: MOMENT-CUR VATURE 311
• F o r w a r d A c t i v a t i o n , ~
e r
k
J
Y l , d ,
y2
Y3 a
I n p u t L a y e r
O u t p u t L a y e r
~1 B a c k w a r d A c t i v a t i o n - -
Fig. 1. Backpropagation network.
t h e a x o n i n t h e h u m a n b r a i n , a n d r e s u l t f r o m
t h e p r e v i o u s p r o c e s s .
T h e o u t p u t t o p r o c e s s i n g e l e m e n t y j, d e t e r m i n e d b y
t h e w e i g h t e d s u m o f t h e i n p u t s i g n a l s ( a s e t o f i n p u t
s ignals , b l ,
b2 .. . ,
b , , ap p l i ed t h r o ug h a s e t o f a s -
s o c i a t ed w e i g h t s , w ~ , w ~ , w ~ , . . . , w j~ , b e c o m e s t h e
i n p u t t o t h e a c t i v a ti o n fu n c t i o n . I n m a t h e m a t i c a l te r m s :
s j = ~ b i w ~ i
i l
y j = f s j ) .
F o r a s i g m o i d ac t i v a t i o n f un c t i o n ,
b e c o m e s :
(1)
(2)
e q u a t i o n ( 2 ) ,
1
y j
=f (sj ) = 1 + e-S/ ' (3)
a n d f o r a h y p e r b o l i c a c ti v a t io n t a n g e n t f u n c t i o n , e q u a -
t i o n ( 2 ) beco m es :
eSj_ e-Sj
y j
= f(s j) = eS + e_Sj (4)
w h e r e
b i = in p u t s i g n a ls t o t h e b a c k p r o p a g a t i o n n e t w o r k ,
y j = o u t p u t s i gn a ls o f t h e b a c k p r o p a g a t i o n n e t w o r k ,
d ~ = d e s i r e d s ig n a ls o f t h e b a c k p r o p a g a t i o n n e t w o r k ,
w ~ = b a c k p r o p a g a t i o n w e i g h t a d j u s t m e n t b e t w e e n
t h e i n p u t an d o u t p u t s i g n a l s .
4 1 Weight correct ion for the output l a y e r
T h e p r o c e ss i n g e l e m e n t s a t t h e o u t p u t l a y e r p r o d u c e
a s i n g l e r e a l n u m b e r f o r e a c h Y l k , Y 2 k , • • . , Y q * ) , w h i c h
a r e t h e n c o m p a r e d t o t h e d e s i r e d o u t p u t d l k , d 2 k , • • . ,
d qg ) t o o b t a i n t h e e r r o r s i g n a l . T h e e r r o r s i g n a l, E q k , is a
m e a s u r e o f th e n e t w o r k ' s p e r f o r m a n c e f o r o n e p r o c e s s -
i n g e l e m e n t in t h e o u t p u t l a y e r , t h a t c a n b e d e t e r m i n e d
f r o m :
E q k = d q k - - Y q k ) ,
(5)
w h e r e
q = a b a c k p r o p a g a t i o n p r o c e ss i n g e l e m e n t in t h e
o u t p u t l a y e r ,
k = r e f e r t o t h e o u t p u t l a y e r i n t h e b a c k p r o p a g a -
t i o n ,
E q k - - - - - a b a c k p r o p a g a t i o n e r r o r s i g n a l f o r o n e p r o c e s s-
i n g e l e m e n t i n t h e o u t p u t l a y e r ;
d q k
= t h e d e s i r e d b a c k p r o p a g a t i o n o u t p u t ,
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312 MAN SOUR NASSER JADID and DAN IEL R. FAIRBA IRN: MOMEN T-CURVATURE
qk - - - - - t h e
a c t ua l b a c k p r o p a g a t i o n o u t p u t f o r o n e p r o -
c e s s ing e l e m e n t .
T he e r r o r va l u e , 6q k, c a n b e ob t a i n e d b y m u l t ip l y i ng
the e r ror s igna l , E q k o b t a i n e d f r o m e q u a t i o n ( 5 ), b y a n
a c t i va t ion f u nc t i on d e r i va t ive :
ofk
6qk= ~sqkEok. (6)
T he e r r o r va l u e , Oq k ob t a i ne d f r om e q u a t i on ( 6 ) , i s
t he n m u l t i p l i e d b y yp j , t he ou t pu t o f one p r oc e s s i ng
e l e m e n t i n t he h i dde n l a ye r , t o p r ov i de t he c onne c t i on
b ( k now n a s t he ge ne r a l i z e d
e i gh t c o r r e c t i on , AWqp k
de l t a r u l e ) . T h i s c o r r e c t e d w e i gh t i s c om pu t e d a s :
A w ~ p.k = r] ~ E q k y p j (V)
w h e r e
Awb e . j
= a d j u s t e d w e i gh t b e t w e e n t he q t h p r oc e s s i ng
e l e m e n t i n t h e o u t p u t l a y e r a n d t h e p t h
p r oc e s s i ng e l e m e n t i n t he h i dde n l a ye r ,
yp~ = b a c k p r opa g a t i on ou t p u t f o r one p r oc e s s i ng
e l e m e n t i n t he h i dde n l a ye r .
4 . 2 . W e i g h t c o r r e c t i o n f o r t h e h i d d e n l a y e r
B e c a u s e o f t he a b s e nc e o f de s i r e d ou t pu t s i n the
h i d d e n l a y e r , t h e p r e v i o u s p r o c e d u r e c a n n o t b e
a do p t e d . T h e r e f o r e , t he e r r o r va l u e , 6p j, f o r t he h i dde n
l a ye r i s ge n e r a t e d w i t hou t t he d e s i r e d ou t pu t s . T h i s is
a c c om pl i s he d b y c a l c u l a t i ng e a c h p r oc e s s i ng e l e m e n t ' s
e r r o r v a l u e i n t h e o u t p u t l a y e r a s o b t a in e d f r o m e q u a -
t ion (6) , 6 q k . T he s e a r e u s e d t o c o r r e c t t he w e i gh t s
g o i n g in t o t h e o u t p u t l a y e r, w h e r e t h e y p r o p a g a t e t o
t he h i dde n l a ye r t o ge ne r a t e 6p j , f o r t he h i dde n l a ye r ,
w h i c h i s c om pu t e d a s :
O f j { ~ - -~W b ~ q k ) .
I n a s i m i l a r w a y , t he h i dde n l a ye r i s c o r r e c t e d b y :
O f j ( + W ~ ] ~ q k)Y o i
w h e r e
A w ~ o . i = t h e
b a c k p r o p a g a t i o n c o r r e c t e d
Y o i
i
8 )
9 )
w e i gh t
b e t w e e n t he p t h p r oc e s s i ng e l e m e n t i n h id -
de n l a ye r a nd t he o t h e l e m e n t i n t he i npu t
l a ye r ,
= t h e b a c k p r o p a g a t i o n o u t p u t f o r o n e p r o c e s s -
i ng e l e m e n t i n t he i npu t l a ye r ,
= r e f e r s to t he i npu t l a ye r i n b a c k p r opa g a t i on ,
= r e f e r s t o t h e h i d d e n l a y e r i n b a c k p r o p a g a -
t ion ,
= a b a c k p r opa ga t i on p r oc e s s i ng e l e m e n t i n t he
inpu t l ayer .
5 . R E Q U I R E M E N T S F O R N E U R A L N E T W O R K
I M P L E M E N T A T I O N S
T he ope r a t i on o f a ne u r a l - ne t w or k t oo l r e q u i r e s t he
s e t ti ng u p o f tr a i n ing a nd t e s t da t a f o r e a c h i nd i v idu a l
t a s k , a nd c o r r e c t pa r a m e t e r s t ha t t he ne t w or k r e q u i r e s
t o p r ov i de a r e a s ona b l e a nd a c c e p t a b l e t r a i ne d
ne t w or k . T he s e c ons i s t o f c o l l e ct i ng da t a , s c a li ng da t a
a nd c hoos i ng t he ne t w or k .
5 . 1 . C o l l e c t i o n o f d a t a
T he da t a i s s e pa r a t e d i n t o t w o s e t s , one f o r t r a i n i ng
a nd t he o t he r f o r t e s t i ng . T he t e s t i ng da t a i s no r m a l l y
t a k e n a s 10% o f t he t r a in i ng d a t a , s u c h t ha t t he 10 th
e l e m e n t o f e a c h t r a i n i ng s e t i s r e s e r ve d f o r t he t e s t i ng
da t a w h i c h w il l p r ov i d e t he b es t p i c t u r e r e p r e s e n t a t i ons
a nd i nc r e a s e t he c on f i de nc e i n t he pe r f o r m a nc e o f t he
t r a i ne d ne t w or k . G e ne r a l l y , t he m or e t r a i n i ng da t a
u s e d , t he b e t t e r t he ne t w or k w i l l pe r f o r m .
5 . 2 . S c a l i n g o f d a t a
T he ne t w o r k a c c e p t s va l u e s on l y f r om 0 t o 1 f o r t he
s i gm oi da l f u nc t i on , a nd - 1 t o 1 f o r t he hype r b o l i c
t a nge n t f u nc t i on . T he p r o c e s s invo l ve s t he c om pu t a t i on
o f l ow a nd h i gh va l u e s o f e a c h t r a i n i ng e x a m pl e da t a
f ield in the selected data f i les .
5 . 3 . C ho i c e o f
n e t w o r k
T h e b a c k p r o p a g a t i o n n e t w o r k u s e s a n o n l i n e a r
r e g r e s s i on t e c hn i q u e t ha t a t t e m p t s t o m i n i m i z e t he
g l ob a l e r r o r , a nd ha s t he a b i l i t y t o p r ov i de c om pa c t
d i s t r ib u t i on r e p r e s e n t a t i ons o f c om pl e x da t a a nd i ts
po t e n t i a l t o m a n i pu l a t e m u l t i p l e - d i m e ns i ona l f u nc -
t ions . Three main aspec t s a re es sent i a l in se lec t ing a
s pe c if ic ne t w or k pa r a d i gm t ha t d i c t a t e s t he c ha r a c t e r i s -
t ic s o f a g ive n ne t w or k . T he s e l e c t ion o f a n a pp r op r i a t e
ne t w o r k i s b a s e d on t he t h r e e f o l low i ng c on f i gu r a t ions :
a r c h i t e c t u r e , t opo l og y a nd ne u r odyna m i c s .
5 .3 .1 . N e tw o rk a rch i tec tu re
B a c k pr opa ga t i on w a s s e l e c t e d f o r t h i s r e s e a r c h
accord ing to the ava i l ab i l i ty of s e r i es pa t t e rn pa i rs ,
w he r e e a c h pa i r c ons i s t s o f a n i npu t pa t t e r n w i t h a
de s i r e d ou t pu t pa t t e r n . T he l e a r n i ng t e c hn i q u e o f f e r e d
b y b a c k p r opa ga t i on i s s u pe r v i s e d l e a r n i ng .
5 . 3 . 2 . N e t w o r k t o p o l o g y
T h e n e t w o r k t o p o l o g y c on s is ts o f th e n u m b e r o f
i npu t a nd ou t p u t l a ye r s , t he nu m b e r o f p r oc e s s ing
e l e m e n t s ( P E s ) t h e y c o n t a in , t h e n u m b e r o f h i d d e n
l a ye r s a nd p r oc e s s i ng e l e m e n t s , t he i r i n t e r c onne c t i v i t y ,
a nd t he p r ope r t i e s o f t he ge om e t r i c a l c on f i gu r a t i ons .
A s a ge ne r a l r u l e, t he a m o u n t o f i npu t da t a t ha t c a n b e
u s e d a s a n u p p e r b o u n d f o r th e n u m b e r o f P E s in t h e
hidden l ayer i s as fo l lows :
R o w ,
UpE Ra ng e * (outpE + ineE) ' (10)
w h e r e
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MANSOUR NASSER JADID and DANIEL R. FAIRBAIRN: MOMENT-CURV ATURE 313
UpE
Row,
OUtpE
inpE
= upper bound for the number of processing
elements in the hidden layer,
= number of rows in training data,
= number of processing elements in the out-
put layer,
= number of processing elements in the input
data,
Range = range between 5 and 10,
PE = number of processing elements.
5.3.3. Netw ork neurodynamics
The generalized delta rule developed by Rumelhart
et al.6
is the most popular learning rule used by back-
propagation networks. Two popular extensions of the
generalized delta rule are implemented in the
NeuralWorks®7 tool, viz the cumulative delta rule
which accumulates weight changes over several exam-
ples, and the normalised cumulative delta rule.
Variants of the backpropagation networks include the
extended delta bar delta (EDBD) , which uses a heuris-
tic and encourages positive learning trends and reduces
oscillation.
5.3.4. Training and testing the network
An iteration process starts after presentation of the
input data with the desired output to the network, and
continues until the network converges to acceptable
levels, or has run for a specified time. The number of
iterations is specified as the number of learnings in the
run menu, or as an acceptable error in the r.m.s.
diagnosis tool. Once the network is trained and con-
verges, the test set is presented to the network sequen-
tially only once, to increase the confidence of the
network performance and account for accuracy.
5.3 .5 . Netw ork per formance
The network's performance is monitored by
root-
mean-square r .m .s .) , weight histogram,
and
confusion
matrix diagnostic instruments provided by the tool to
achieve a better understanding of the network's perfor-
mance. The r .m.s , error is computed from: 8
where
?p~t E (do.t-YouO:
oPE
r . m . s .
nptnoPE
(11)
r.m.s. = root-mean-square,
pt
= summation over all patterns in the training
set,
=summation over all output processing ele-
oPE
ments,
dour = desi red output,
Yout =
network output,
npt ----
number of patterns in the training set,
hoPE
- - - - n u m b e r
of processing elements in the out-
put layer.
The weight histogram provides a normalized histo-
gram of all the variables in the network that change
during the training session that is used to monitor the
network's performance. The x-axis along the confusion
matrix
provides the network output, and the y-axis is
the desired output. The interior quadrants are discre-
tized into bins to show the network outputs. A value of
one means an excellent correlation between the desired
and network outputs. The
epoch size
is one o f the main
factors that control the convergence of the network,
where weight changes in the network are accumulated.
The epoch size can be tuned and adjusted to provide
better learning procedures by monitoring it as it
evolves.
6 R E L A T I O N S H IP B E T W E E N M O M E N T A N D
C U R V A T U R E
Previous experimental work was carried out by
Nirjar9 to investigate the structural behaviour of cast-
in-situ beam-column joints under static loading con-
ditions. The study investigates the relationship between
the behaviour of beam-column joints and geometrical
shape, amount and size of steel reinforcement, fixed
beam and column cross-sectional dimensions and con-
crete strength. Tests were carried out on a total of 34
specimens under the following conditions:
1. Series NN is based on variations of column
loading conditions, from 10 to 60 of the
ultimate column load.
2. Series NM is based on variations of column
longitudinal reinforcement, Pc, tested at 10 of
the ultimate load loading.
3. Series NO is based on variations of column
longitudinal reinforcement, Pc, tested at 50 of
the ultimate column loading.
4. Series NP is based on variations in the area of
tensile reinforcement in the beams, tested at
10 of the ultimate column load.
5. Series NQ had the same reinforcement varia-
tions as the NP series, but tested at 60 of the
ultimate column load.
6. Series NR involved variations in the area of
transverse reinforcement in the column and
joint, tested at 10 of the ultimate load.
7. Series NS had variations in the area of lateral
reinforcement in the beams using different
spacing of stirrups.
8. Series NT covered five dif ferent concrete
grades, viz. 20, 30, 35, 40 and 45 N/mmz.
The research work presented is part of a complete
and comprehensive investigation into the behaviour of
corner beam-column joints under biaxial bending
moments. The work carried out follows previous exper-
imental work done by Nirjar 9 into the investigation of
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314 MANSOURNASSERJAD ID and DAN IEL R. FA IRBAIRN:MOMENT-CURVATURE
34 s pe c i m e ns . T he p r i m a r y w or k de s c r i b e d he r e i s on l y
a pp l i e d t o i nve s t i ga t e t he m om e n t - c u r va t u r e r e l a t i on -
s h i p fo r t he f le x u r al m e m b e r . A p a r t o f t he s t ra t e gy
i m p l e m e n t e d t o c a r r y ou t th i s r e se a r c h w or k i s s how n i n
F i g . 2 , u nde r t he p r e pa r a t o r y s t a ge .
A u n i a x i a l l o a d , m o m e n t a n d c u r v a t u r e p r o c e d u r e
I
I a s e
o n
E x p e r i m e n t
R e c e n t R e s e a r c h
N e w C o d e o f P r a c t ic e
N o
, [Num.r ic l l N.u r l l ~ . /
R ~ ~w S_~gy_. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .__~___.
I
o
a s e o n R e c e n tR e s e a r c h
NewCodeofVrectice ~
~Numerical NN m l
/ No~o,-bSas. I"
I I M IM o d e l I
s*op 3
Fig. 2. Strategies or the implementationof predictive stages.
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M A N S O U R N A S S E R J A D I D a n d D A N I E L R . F A I R B A IR N : M O M E N T - C U R V A T U R E
p M
p M ~ . . . . . . . . . . . . . . . . . . . . . . . . . . c
e,
o a o, (a) Load De ~,nf l~ 4 , L , # , # , ÷
( b ) M o m e n t - C u r v a t u re
315
F i g . 3 . Id e a l i z e d l o a d - d e f l e c ti o n a n d m o m e n t - c u r v a t u r e r e l a t io n s h i p s .
w a s d e v e l o p e d b y P f a n g
et al. 1°
T h e y d e s c r i b e d a
p r o c e d u r e t o e v a l u a t e t h e r e l a t io n s h i p b e t w e e n u n i a x ia l
l o a d , m o m e n t a n d c u r v a t u r e b a s e d o n u s i n g a t o o l f o r
t he a na l y t i c a l p r oc e du r e . A s i m i l a r p r oc e du r e w a s
a d o p t e d b y K r o e n k e et al . u w h o a l s o c o n s i d e r e d t h e
e f f e c t s o f a s y m m e t r i c a l r e i n f o r c e m e n t p l a c e m e n t a n d
t he i nc l u si on o f s t r a in ha r de n i ng . A n i nc r e m e n t a l - t ype
m e t h o d f o r t h e d e t e r m i n a t i o n o f t h e l o a d - d e f o r m a t i o n
r e l a t i ons h i p w a s de ve l op e d b y E I - M e t w a l l y a nd C he n .12
T h i s m e t h o d h a d b e e n e x p e r i m e n t a l l y v e r if ie d , a n d c a n
b e a p p l i e d t o a no n l i ne a r a na l y s i s o f f a m e s t r u c t u r e s .
A r e l a t io n s h i p b e t w e e n l o a d - d e f o r m a t i o n a n d
m o m e n t - c u r v a t u r e c a n b e d e t e r m i n e d b y a p p ly i n g t h e
p r i nc i p l e o f e q u i l i b r i u m o f t he i n t e r na l f o r c e s a nd
c om pa t i b i l i t y o f t he s t r a in s . T he m e m b e r u s u a l l y u nde r -
g o e s t h r e e s t a g e s o f b e h a v i o u r d u r i n g a p p l i c a t io n o f t h e
l oa d . F i gu r e 3 s how s t yp i c a l
i d e a l i z e d
l oa d - de f l e c t i on
a n d m o m e n t - c u r v a t u r e r e la t io n s h i p s.
T he l oa d i ng s t a ge s a r e c ha r a c t e r i z e d b y :
( a ) A n e l a s t i c s t a ge b e t w e e n 0 a nd a : t h i s s t a ge i s
u s u a l l y r e f e r r e d t o a s t h e " u n e r a c k e d s t a g e " .
T he t e ns i l e r e i n f o r c e m e n t a t t h i s s t a ge i s b a s i -
ca l ly inac t ive .
( b ) T h e y i e ld s ta g e b e t w e e n a a n d b .
( c ) T he u l t i m a t e s t a ge b e t w e e n b a nd c .
6 1 Y i e ld m o m e n t a n d c u r v a t u r e
A s t he l oa d i nc r e a s e s , t he c r a c k s f o r m i ng i n t he
t e n s i o n z o n e p r o p a g a t e u p w a r d t o w a r d t h e n e u t r a l a x is .
T he c onc r e t e i n t he t e ns i on z one i s i na c t i ve , a nd t he
t e ns i l e s t e e l r e s i st s t he e n t i r e t e ns i on . A s i l l u s t r a t e d i n
F i g . 3 ( b ) o f t he m om e n t - c u r va t u r e r e l a t i ons h i p , b is t he
po i n t on t he c u r v e w h i c h de f i ne s y i e l d i ng o f t he t e ns i on
r e i n f o r c e m e n t w i t h f u r t he r i nc r e a s e i n t he de f l e c t i on ,
w h i le t h e a p p l i e d l o a d r e m a i n s n e a r l y c o n s t an t . A s l o n g
a s t he c r o s s - s e c t i on i s u nde r - r e i n f o r c e d , t he t e ns i l e
r e i n f o r c e m e n t r e a c h e s y i e l d b e f o r e t h e c o n c r e t e
c r u s he s . T he y i e l d c u r va t u r e , ~ y , ge ne r a l l y de f i ne d a s
t he c u r va t u r e a t w h i c h t he t e ns i l e r e i n f o r c e m e n t
r e a c he s i t s y i e l d po i n t s t r e s s , i s de t e r m i ne d f r om F i g .
4 ( b ) b y s t r a in c om p a t i b i l it y a s f o l low s :
gy fy
~ Y = d - x d - Es(1 - x ) d (12)
F o r a d o u b l y r e i n f o r c e d b e a m w i t h a t e n si le r e i n fo r c e -
m e n t r a t i o o f :
AS
P = b--d; (1 3)
a nd c om pr e s s i ve r e i n f o r c e m e n t r a t i o o f :
AS'
p ' = - - ( 14 )
b d
t he f o l l ow i ng e x p r e s s i on c a n b e de r i ve d :
x = X /(p + p' )2n2 +
2 n ( p + p ' a ) - n ( a + p ' )
w h e r e
( 15 )
E~
n = mod ula r ra t io , equ a l to ~ -~,
E s - - m o du l u s o f e l a s ti c i ty o f s t e e l , N / m m 2,
d = e f f e c ti ve de p t h o f t he c r o s s - s e c t i on , m m ,
a d
= d i s t a n c e f r o m t h e c o m p r e s s i o n f a c e o f m e m b e r
t o t he c e n t r i od o f c om pr e s s i on s t e e l , r a m ,
A s = a r e a o f t e ns i on s t e e l r e i n f o r c e m e n t , m m 2 ,
A ¢ = a r e a o f c om p r e s s i on s t e e l r e i n f o r c e m e n t , m m 2 ,
/ 9= t e ns i le r e i n f o r c e m e n t r a t i o o f s t e e l ,
p ' = c om pr e s s i ve r e i n f o r c e m e n t r a t i o o f s t e e l .
T he f ina l e x p r e s s ion f o r t he c u r va t u r e a t y i e l d , ~ y , is
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316 MANSOUR NASSER JADID and DANIEL R. FAIRBAIRN: MOMENT-CURVATURE
t h e n e s t a b l i s h e d b y c o m b i n i n g e q u a t i o n s ( 1 2 ) a n d ( 1 5 )
t o o b t a i n :
Cpy = E s [ n ( p + p ) - V ( p + p ) Z n 2 + 2 n ( p + p a ) + 1 ]d '
(16)
T h e s t r a i n i n th e c o m p r e s s i o n s t e e l i s d e t e r m i n e d f r o m
Fig . 4(b) as :
es, = ( xd - ad )~Oy = d (x - a ) ¢ y .
W h e r e t h e c o m p r e s s i o n s t e e l d o e s n o t y i e l d , i . e .
es, < e y ,, th e c o m p r e s s i v e f o r c e d u e t o t h e c o m p r e s s i o n
steel is C~, where:
C~, = e~,E e4s,.
W here the com press ion s tee l y ie ld s , i . e . e s , -> E y , , t h e
c o m p r e s s i v e f o r c e d u e t o t h e c o m p r e s s i o n s t e e l i s C , ,
w h e r e :
¢~ =As L .
A p p l y i n g i n t e r n a l e q u i l i b r i u m :
T = C c + C ¢
so tha t :
C c = T - C s , = A s f y - A s , f , , .
T he in te rna l r es i s t ing m om en t o f the c r i t ica l c ro ss -
s e c t i o n i s t h e r e f o r e o b t a i n e d b y t a k i ng t h e m o m e n t s o f
t h e i n t e r n a l f o r c e s o f t h e c o m p r e s s i v e c o n c r e t e a n d t h e
c o m p r e s s i v e r e i n f o r c e m e n t a b o u t t h e t e n s i l e r e i n f o r c e -
m e n t :
Subs t i tu t ing fo r C e and C s, in to eq ua t i on (22 ) , the
y i e ld m o m e n t , My,is g iven as:
M y = d [ ( Z s f y - a s , f s , ) ( 1 - 3 ) + Z s , f s , ( 1 - a ) ] . ( 2 3 )
T h e s t r e s s i n t h e c o m p r e s s i o n r e i n f o r c e m e n t , f s , , c a n
b e o b t a i n e d f r o m F i g . 4 , a n d f r o m t h e c o m p r e s s i o n
s t r a in ob ta ined ear l ie r , a s :
( x - a ) (24)
(17 ) f~, = Eses, = Esey (1 - x ) '
w h e n t h e s t r e s s i n t h e c o m p r e s s i o n r e i n f o r c e m e n t
r eac hes the y ie ld s t r es s ( f~ , = fy , ) and
C~, =f~,A~,. (25 )
( 1 8) P r e v i o u s c o m p u t a t i o n s p r o v i d e d e x p r e s s i o n s w h i c h
r e l a t e t h e c u r v a t u r e a n d m o m e n t a t t h e y i e l d s t a g e f o r a
d o u b l y r e i n fo r c e d b e a m .
F o r a s in g l y r e i n f o r c e d b e a m , t h e e x p r e s s i o n f o r r a t i o
x can be s im p l i f ied to :
(19) x = X / ( p 2 n z + 2 n p ) - n p . (26)
T h e d i s t a n c e o f t h e n e u t r a l a x i s f r o m t h e c o m p r e s s i v e
f a c e b e c o m e s :
(20)
x d
= (X/(p2n 2 +
2 n p ) - n p ) d .
(27)
T he cu rv a tu re a t y ie ld i s s im p l i f ied to :
L
(21) q~Y E s [ n p - X /((p Zn 2 + 2 n p ) + 1 ]d ' ( 28 )
a n d t h e m o m e n t f o r a s i n g l y r e i n f o r c e d b e a m b e c o m e s :
(22)
6 .2 . P r e par ator y s tage for y i e ld mom e nt and c ur vatur e
F o r th e p r e p a r a t o r y s t a ge , t h e p r o c e d u r e a d o p t e d i n
t h i s s t u d y t o d e t e r m i n e t h e m o m e n t c u r v a t u r e r e l a t i o n -
d
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
a) Cr oss sectio n CO) Stra ins
(C) elastic stress dis tr ibution
Fig. 4. Elastic stress distribution condition at yielding of the tensile reinforcement.
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M A N S O U R N A S S E R J A D ID a nd D A N I E L R . F A I R B A I R N : M O M E N T - C U R V A T U R E
T a b l e 1 . C o m p a r i s o n s o f m o m e n t s a n d c u r v a t u r e s at y i e l d b y N i rj a r 9 a n d b y J a d i d5 for the preparatory s tage
317
S p e c i m e n s
N i r jar 9 N e u r a l n e t w o r k s5
E r r o r = A b s ( 1 N e t w o r k ~ • 1 0 0
\
( N ) ( N )
q~y
* 10 - s My My C y * 10 - s My (N) w . r . t ,
w.r.t .
f'~ p
C o m p u t e d C o m p u t e d E x p e r i m e n t a l
( N ) ( N ) ( C ) ( C ) ( E )
( N / m m 2 ) ( % ) ( C ) ( C ) ( E ) ( m m - ~ ) ( k N m m ) C y M y M y
NN1 30 .0 1 .28 2120 2829 2880 2062 .99 2615 .82 2 .69 7 .54 9 .17
NN2 30 .0 1 .28 2120 2829 294 0 2062 .99 2615 .82 2 .69 7 .54 11 .03
NN~ 30 .0 1 .28 2120 2829 2850 2062 .99 2615 .82 2 .69 7 .54 8 .22
NN4 30 .0 1 .28 2120 2829 2790 2062 .99 2615 .82 2 .69 7 .54 6 .24
NNs 30 .0 1 .28 2120 2829 2760 2062 .99 2615 .82 2 .69 7 .54 5 .22
NN 6 30 .0 1 .28 2120 2829 2760 2062 .99 2615 .82 2 .69 7 .54 5 .22
NM7 30 .0 1 .28 2120 2829 2880 2062 .99 2615 .82 2 .69 7 .54 9 .17
NM s 30 ,0 1 .28 2120 2829 2880 2062 .99 2615 .82 2 .69 7 .54 8 .22
NM9 30 ,0 1 .28 2120 2829 2820 2062 .99 2615 .82 2 .69 7 .54 7 .24
NM10 30 .0 1 .28 2120 2829 2820 2062 .99 2615 .82 2 .69 7 .54 7 .24
NO n 30 ,0 1 .28 2120 2829 2760 2062 .99 2615 .82 2 .69 7 .54 5 .22
NO l 2 30 ,0 1 .28 2120 2 829 2760 2062 .99 2615 .82 2 .69 7 .54 5 .22
NO13 30 .0 1 .28 2120 2829 2760 2062 .99 2615 .82 2 .69 7 .54 5 .22
NO t 4 30 .0 1 .28 2120 2829 2910 2062 .99 2615 .82 2 .69 7 .54 10 .11
NPI5 30.0 0.72 1900 1637 1650 1881.73 1632.30 0.96 0.29 1.07
NPI6 30 .0 2 .00 2360 4 310 4 200 24 4 5 .08 5067 .66 3 .60 17 .58 20 .66
NPI7 30 .0 2 .55 254 0 4 312 5260 2563 .34 5870 .79 0 .92 36 .15 11 .61
N P l s
30 .0 2 .99 2670 6281 5760 2595 .78 6105 .05 2 .78 2 .80 5 .99
NQ19 30 .0 0 .72 1900 1637 1680 188 1 .73 1632 .30 0 .96 0 .29 2 .84
NQ~0 30 .0 2 .00 2360 4 310 4 200 24 4 5 .08 5067 .66 3 .60 17 .58 20 .66
NQ21 30 .0 2 .55 254 0 54 12 5160 2563 .34 5870 .79 0 .92 8 .4 8 13 .78
NQ22 30 .0 2 .99 2670 6281 5690 2595 .78 6105 .05 2 .78 2 .80 7 .29
NR 23 30 .0 2 .99 2670 6281 5880 2589 .51 6057 .88 3 .01 3 .55 3 .03
NR 24 30 .0 2 .99 2670 6281 5820 2589 .51 6057 .88 3 .01 3 .55 4 .09
NR 25 30 .0 2 .99 2670 6281 5760 2589 .51 6057 .88 3 .01 3 .55 5 .17
NR 26 30 .0 2 .99 2670 6281 5760 2589 .51 6057 .88 3 .01 3 .55 5 .17
NS27 30 .0 1 .28 2120 282 9 2700 2062 .99 2615 .82 2 .69 7 .54 3 .12
NS28 30 .0 1 .28 2120 2829 2730 2062 .99 2615 .82 2 .69 7 .54 4 .18
NS ~ 30 .0 1 .28 2120 2829 2700 2062 .99 2615 .82 2 .69 7 .54 3 .12
NS~0 30 .0 1 .28 2120 2829 2760 2062 .99 2615 .82 2 .69 7 .54 5 ,22
NT31 4 0 1 .28 204 2857 2910 1985 .51 264 1 ,02 2 .67 7 .56 9 .24
NT32 35 1 .28 2070 284 6 2850 2027 .80 2623 .67 2 .04 7 .81 7 ,94
NT33 25 1 .28 2170 2812 2790 2107 .50 2607 .58 2 .88 7 .27 6 .54
NT34 20 1 .28 2250 2790 2700 2161 .82 2623 .14 3 .92 5 .98 2 .85
sh ip i s to se l e c t a ne twor k o f (6 ,40 ,30 ,2 ) . Thi s r e pr e -
s e n t s t h e n u m b e r o f i n p u t P E s , t h e h i d d e n P E s i n t h e
f i r s t l aye r , the h idde n P Es in the se c ond h idde n laye r
a n d t h e P E s i n t h e o u t p u t l a y e r , r e s p e c t i v e l y . T h e
r .m.s , wa s se t to 0 .02 , and pr ov ide d sa ti s factor y
r e s u l t s . T h e E D B D l e a r n i n g r u l e a d o p t e d i n c l u d e d a
h e u r i st i c a d j u s t m e n t t o t h e m o m e n t u m t e r m , s e t t o 0 . 4 .
The l e ar ning r a te s we r e se t to 0 .3 for the f i r s t h idde n
laye r , 0 .25 for the se c ond h idde n laye r and 0 .15 for the
output l aye r . Input and de s i r e d pat te r ns we r e pr e -
se nte d to the ne twor k r andomly in the t r a in ing f i l e and
se que nt ia l l y in the t e s t f i l e . The t r a in ing and te s t f i l e s
w e r e g e n e r a t e d w i t h t h e F O R T R A N p r o g r a m b y
s e l e c t i n g a p o s s i b l e c o m b i n a t i o n o f 3 4 s p e c i m e n s .
Th r e e c onc r e te c y l inde r s tr e ngths , 20 , 30 and 45 N/mm :
we r e use d , wi th two d i f f e r e nt bar ar r ange me nts , 2 and
4 , as we l l a s two d i f f e r e nt bar d iame te r s , 6 and 12 mm ,
t o g e n e r a t e t h e d a t a . T h e t o t a l n u m b e r o f c o m b i n a t i o n s
o b t a i n e d w a s t h e r e f o r e 4 0 8 , o f w h i c h o n l y 2 4 3 p a t t e rn s
we r e use d for t r a in ing and 29 pat te r ns for the t e s t ing
f i l e . T h e r e m a i n d e r d i d n o t c o n f o r m t o t h e C o d e o f
P r ac t i c e . The pr e l iminar y t r a in ing and te s t pat te r n
va lue s obta ine d fr om a nume r ic a l ana lys i s for the c ur -
vatur e , ~y , w e r e ve r y smal l , a s obta ine d fr om us ing the
F O R T R A N p r o g r a m . T h e r e f o r e , t o e n h a n c e t h e
n e t w o r k p e r f o r m a n c e , t h e v a l u e s o f t h e c u r v a tu r e , ~ y ,
we r e magn i f ie d by 08 and the s t e e l p e r c e ntage , p , b y 04 .
T h i s p r o v i d e d a b e t t e r n e t w o r k p e r f o r m a n c e . T h e
r e sul t s pr e dic te d by the ne ur a l ne twor ks for the y i e ld
c ur vatur e s and mome nts ar e in c lose agr e e me nt wi th
those obta ine d by N ir jar . 9 Th e se r e sul ts ar e tabu la te d
in Table 1 , toge the r wi th the pe r c e ntage e r r or s .
T h e p r e d i ct e d f o r m u l a e o b t a i n e d b y n e u r a l n e t w o r k s
for the c ur vatur e c an be pr e se nte d in the for m of :
your.p _ ~out.p (478 .41) + 2206 .8 (30 )
78y ~'~7 8, y
x78.ut.~p _- tanh(Z x~s, y . (31 )
T h e m o m e n t a t y i e ld o b t a i n e d c a n b e e x p r e s s e d a s:
Y 7 9 , ° u t y
_- x79.°ut'y . (2 8 2 4 .0 5 6 ) + 3 8 7 7 .1 9 5 (32)
X79,oUty ----
an h (,~r~9. y
(33)
in w hic h out, . . . . t ,p
78,y
and . ,79,y are the scaled curvature and
mome nt a t y i e ld , r e spe c t ive ly , for the pr e par ator y
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3 18 M A N S O U R N A S S E R J A D I D a n d D A N I E L R . F A I R B A I R N : M O M E N T - C U R V A T U R E
r t m u m t - C u e v a t u r e Ro a t Ion ; C~c o=4 74?9 ; NSS :. _em~__; r51 :1. ; ~ : 1. ; cR : 1.15
j
j L . . . = _ I
id ro t ~ lbt't, l~
2
. . h . . . I
I C o n .
l ~ t a - l x
1
a ) Tr a ine d Ne t wo r k .
r l m e o n ~ C u ~ a t u e e R e l a t t o n ; C v c 1 o = 47 4 79 ; N ~ : . ~ ; C r 1 :1 . ; C R : . 8 ; T e e t = 2 9
IC, nf . Ma t.rlx I
I c o n e M a t r i x 2
b) Tr a ine d a nd Te s t e d Ne t wo r k .
F i g. 5 . C u r v a t u r e a n d m o m e n t a t y i el d f o r t h e p r e p a r a t o r y s t a g e . 7
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MA N SOUR N A SSE R JA DID and DA N IE L R . F A IRB A IRN : MOME N T- C URV A TURE 319
s t a g e . T h e n e t w o r k a r c h i t e c t u re , t o p o l o g y a n d n e u r o -
d y n a m i c s a r e s h o w n i n F i g . 5 f o r t h e t r a i n e d a n d t e s t
n e t w o r k s . T h e n e t w o r k c e a s e d l e a r n i n g a t a 4 7 , 4 7 9
c y c l e s, w i t h e x c e l l e n t c o r r e l a t i o n s o f 1 .0 f o r b o t h t h e
t r a i n e d a n d t e s t n e t w o r k s i n t h e
c o n f u s i o n m a t r i c e s .
T h e c las s i f i ca t ion ra te w a s 1 . 0 , w h i c h i n d i c a t e s t h a t t h e
n e t w o r k c o r r e c t l y c la s si fi e d 1 0 0 % o f th e g o o d c a t e -
g o r i e s a n d d i d n o t m i s - c la s s if y a n y o t h e r c a t e g o r i e s .
T h u s , t h e c u r v a t u r e a n d m o m e n t f o r a re i n f o rc e d
b e a m a t th e y i e ld s t a ge c a n b e c o m p u t e d f r o m e q u a -
t i o n s ( 1 6 ) a n d ( 2 3) r e s p e c t i v e l y , a s p r o p o s e d b y N i r j a r , 9
o r a l t e r n a t iv e l y b y th e n e u r a l n e t w o r k a s p r o p o s e d b y
e q u a t i o n s ( 3 0 ) a n d ( 3 2 ) ,
7 . C O N C L U S I O N
N e u r a l n e t w o r k s c a n p r o v i d e a n a l t e r n a t i v e t o c o n -
v e n t i o n a l m e t h o d s b y p r o v i d i n g a n i n s id e r e l a t io n s h i p
in t h e f o r m o f g e n e ra l iz a t io n s b e t w e e n t h e p a r a m e t e r s
i n v o l v e d . T h e r o l e o f n e u r a l n e t w o r k s i n e x p e r i m e n t a l
i n v e s t ig a t i o n s h a s b e e n d i s c u s s e d , a n d t h e a u t h o r s h a v e
d e m o n s t r a t e d t h e i r p o t e n t i a l i t y i n ass i s t ing e x p e r i m e n -
t al w o r k . T h e p r i m a r y o b j e c t iv e w a s d e m o n s t r a t e d b y
s t u d y in g f e w e r t e st s p e c i m e n s a n d m a p s o f n -
d i m e n s i o n a l s p a c e t o t r a c k t h e r e m a i n i n g d a t a , w h i c h
u l t i m a t e l y r e su l t s i n re d u c i n g t h e n u m b e r o f s p e c i m e n s
t e s t e d , w i t h c o n s e q u e n t e c o n o m i c b e n e f i t s f o r th e
e x p e r i m e n t a l w o r k . T h e
c o n c e p t
a n d
m e t h o d o l o g y
c a n
b e a l s o i m p l e m e n t e d t o p e r f o r m a d d i t i o na l p r e l i m i n a r y
e x p e r i m e n t a l w o r k t o p r o v i d e i n f o r m a t i o n o n t h e r e la -
t i o n s h i p s b e t w e e n t h e m a t e r i a l s , l o a d i n g a n d g e o m e t r i -
c a l s h a p e .
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A U T H O R S B I O G R A P H I E S
D r J a d i d w a s a w a r d e d a B S C E ( C iv i l E n g i n e e r in g ) d e g r e e f r o m t h e U n i v e rs i ty o f W a s h i n g to n , S e a t tl e i n 1 9 81 a n d a n M S C E
( St r u c t u r a l E ng i nee r i ng) degr e e f r om Pennsy l van i a S t a t e Uni ve r s i t y in 1987 . In 1994 , he su ccess fu ll y com pl e t ed h i s Ph .D .
t hes i s a t E d i nb u r gh Uni ve r s i t y . H e i s cu r r en t l y an A ssoc i a t e Pr o fe s so r a t Ki ng F a i s a l Un i ve r s i t y , Sau d i A r ab i a . Rec en t j o i n t
pu b l i ca t i ons have b een i n Artificial Intelligence fo r E ngineering Desig n, a n d Expert Sys tems with Applicat ions .
D r F a i r b a i r n is D e p u t y H e a d o f th e D e p a r t m e n t o f C i v il a n d E n v i r o n m e n t a l E n g i n e e r i n g a t t h e U n i v e r s it y o f E d i n b u r g h . P a s t
pu b l i ca t i ons have i nc l u ded pap e r s i n t he A 71 Structural Journal, Magazine of Concrete Research a n d Ma sonry International.